cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A019584 a(n) = n^2*(n-1)^3/4.

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%I A019584 #36 Jul 08 2025 07:32:34
%S A019584 0,0,1,18,108,400,1125,2646,5488,10368,18225,30250,47916,73008,107653,
%T A019584 154350,216000,295936,397953,526338,685900,882000,1120581,1408198,
%U A019584 1752048,2160000,2640625,3203226,3857868,4615408,5487525,6486750,7626496,8921088,10385793
%N A019584 a(n) = n^2*(n-1)^3/4.
%H A019584 Vincenzo Librandi, <a href="/A019584/b019584.txt">Table of n, a(n) for n = 0..600</a>
%H A019584 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A019584 a(n) = Sum_{j=1..n-2} Sum_{i=1..n-2} (i^3 + j^3)/2. - _Alexander Adamchuk_, Oct 24 2004
%F A019584 G.f.: x^2*(1 + 12*x + 15*x^2 + 2*x^3)/(1 - x)^6. - _Colin Barker_, May 04 2012
%F A019584 a(n) = Sum_{i=0..n-1} (n-1)*(n-1-i)^3 for n>0. - _Bruno Berselli_, Oct 31 2017
%F A019584 From _Amiram Eldar_, Feb 13 2023: (Start)
%F A019584 a(n) = A099903(n-1)/2.
%F A019584 Sum_{n>=2} 1/a(n) = 16 - 2*Pi^2 + 4*zeta(3).
%F A019584 Sum_{n>=2} (-1)^n/a(n) = 24*log(2) - 16 - Pi^2/3 + 3*zeta(3). (End)
%t A019584 Table[n^2*(n-1)^3/4, {n,0,100}]
%o A019584 (Magma) [n^2*(n-1)^3/4: n in [0..60]]; // _Vincenzo Librandi_, Apr 26 2011
%o A019584 (PARI) a(n)=n^2*(n-1)^3/4 \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A019584 Cf. A099903.
%K A019584 nonn,easy
%O A019584 0,4
%A A019584 _N. J. A. Sloane_